What harm could deficits do to the economy?
When the demand for borrowing increases (especially by the government), does that increase rates or not?
Borrowing demand usually follows rates, when rates are low private organizations borrow...because it's cheaper to borrow. Governments don't have that luxury of decision making or use for the borrowed money. It feels like the Government's need to borrow does nothing directly to rates.
Interest rates are driven by the Fed, as policy. They raise rates if they want to slow down the economy. They lower rates if they want to speed up the economy. The latter results in more borrowing (and therefore investment) by the private sector.
Summary: I don't think there's a causal relationship between budget deficits and interest rate movements (up or down).